Do
you still have nagging questions in your mind about lenses, aperatures
and exposure? Are you confused about circles of confusion?
You might be surprised to discover that even many published textbooks
present misleadingly kooky
information on the subject of basic lens optics.

Would
you like to have a more intuitive grasp of how a lens works?

Try
this simple thought experiment:

Imagine
two points in space, Point A on a brightly colored
object, and Point B, on a white wall.

Point
B receives light rays from billions of rays coming from billions
of places. Its final color is the sum total of all of the incoming light
from each ray.

Point
A is a point on a well lit, brightly colored object. (In this
case, a bright red chili pepper.) It gives off rays of red light in
billions of different directions, but only one of those billions of
rays travels from Point A to Point B. Because (in our illustration)
only one of the billions of rays reaching Point B is red, there is no
chance for a human observer to perceive Point B as having any sort of
reddish color to it.

Suppose
we could position a lens between Point A and Point
B in such a way as to take millions of red rays from Point
A — rays never originally destined for Point
B — and redirect them so that they could contribute their
light to Point B.

Point
B still receives billions of rays of light, but now, thanks
to the lens, millions of those rays come from Point A.
You can imagine that with those proportions in effect, Point
B now takes on a reddish color.

Since
the same can be said for any point on the object, and since any adjacent
points on the object "map" to adjacent (but upside-down and
backwards) locations on the screen, the lens is "projecting"
an image of the object onto the screen.

Let's
set up an aparatus to test this theory.

The
lens projects an image of an object onto a screen by gathering rays
"never intended for the screen" and bending them back towards
the screen until they arrive in such quantities at the surface of the
screen that they "outnumber" the other rays coming from all
kinds of other sources.

(You
can see in my aparatus that I've used a square sheet of opaque cardboard
to block many ambient rays, thereby helping the lens rays outnumber
the ambient rays, resulting in an apparently brighter projected image.
In creating a front and back wall, I'm well on my way toward creating
a sealed, light-tight box — also known as a camera.)

An
iris reduces the amount of light passing through a lens by reducing
the size of the aperature through which that light can pass. Make the
lens half as big in area, and half of the rays that were redirected
by the lens from Point A to Point B
now resume their original paths. They miss Point B
completely, and Point B receives half the light it
was getting with the larger lens.

And
here's the fun part: It dosn't matter what you use to reduce the size
of the lens. You can stop down with a fancy many-bladed
iris mechanism, or you can even use your hand to cover
half of the lens. You won't lose half the picture, just half the light.
(By the way, losing half of the light through a lens is defined as losing
one f-stop of exposure)

The
lens makes all of the rays from Point A converge at Point
B. If my lens is round then the incoming cone of rays aimed at
Point B has a cricular cross section. If I move my movie screen toward
or away from the lens, I'll intercept that cone of incoming rays at a
place other than its apex. This will cause my image to go out
of focus.

When
you look at a blurry photograph, you can actually see the shape of the
lens by looking at the bokeh — the shape into
which pinpoints of light are spread by the lens of the camera.

Remember,
the bundle of rays that travels from object to lens forms an outgoing
cone. That same bundle is gathered by the lens and reshaped into an
incoming cone whose apex lies in the plane of the screen. Move the screen
toward or away from the lens such that you intercept that cone elswhere,
and you form a blurry picture in which each point of light from the
source object becomes one of many overlapping circles. These circles
are what photographers refer to when they discuss bokeh.

While
(in a blurry projection) every point from the source image "maps
to" a circle, a photograph's bokeh becomes most noticeable in the
highlights of the image. It bears very little resemblance to the gaussian
blur so often used by digital compositors
to suggest shallow depth of field. If the lens itself is not circular
(because of the shape of the aperature) then neither will be the bokeh.
Additional complications or refinements in the optics of a camera can
change the distribution of light intensity across its bokeh. Some people
render CG which exhibits bokeh.

Small
lenses have small bokeh. They also gather less light, which is why they
require that brighter light be supplied to the environment, or sometimes
that what light there is be allowed to accumulate (on a recording medium
like film) over a longer period of exposure time.

A
pinhole camera lens, the smallest lens of all, has
virtually no bokeh. It also gathers almost no light, in theory allowing
only one ray (in the context of our illustration above) to pass from
Point A to Point B. Although it contains
no glass, a pinhole camera lens does not operate in a way that is qualitatively
different from the way in which a glass lens works. Both kinds of lenses
have this in common: they both convey to Point B only
that light which originates from Point A, and from
no other source. Because a pinhole camera lens operates with so little
light, it cannot allow to pass through itself a quantity of rays sufficient
to compete with much ambient light at all, and if it is to project a
discernable image onto film, it must operate in a light-tight camera
and with relatively long exposure times.

I
took these pictures with a commercially
available high precision wooden pinhole camera, but if you're so
inclined, you can make your own simple pinhole camera without too much
difficulty. The behavior of a typical CG lens most closely resembles
that of a pinhole camera lens.

According
to a common misconception, image distortion results from the kind of lens
one uses. In many people's minds, a wide-angle lens produces
great distortion, while a telephoto lens "flattens
images."

In
fact the distortion arises from the distance between camera
and object, and it becomes most apparent when a distantly-photographed
object is unnaturally magnified by a telephoto lens, or when a closely-photographed
object is unnaturally reduced by a wide-angle lens.

In
the diagrams above, three unit cubes were simultaneously rendered at
4K resolution using a (very wide-angle) 10mm lens on Maya's default
film back. All three cubes share common vanishing points (look closely).

The
distant cube seems "flatter" than the others because the relative
size difference between the front cube face, 100 units away, and the
rear cube face, 101 units away, is small. The front cube, by contrast,
seems greatly distorted because the front face of the unit cube, at
1 unit away, is twice as close to the camera as is the rear face of
the cube, at 2 units away.

Close
objects display distortion by virtue of their closeness. People
associate distortion with wide-angle lenses because such lenses are
commonly used to pleasingly frame close objects. The wide angle
lens is not the cause of the distortion.

Distant
objects display flatness by virtue of their distance. People
associate flatness with telephoto lenses because such lenses are commonly
used to pleasingly frame distant objects. The telephoto lens
is not the cause of the flatness.